# Equivalent fractions

Equivalent fractions are fractions that, although they have different digits in the numerator and denominator, represent the same value. For example, the fractions , , and all represent the same part of a whole: 1/2. This can be depicted using fraction bars:

Even though each of the fraction bars is broken up into a different number of parts, each of the fraction bars represents the same fraction, albeit with different numerators and denominators.

Any given fraction has an infinite number of equivalent fractions. We can find different equivalent fractions by multiplying the numerator and denominator of a given fraction by a fractional representation of 1. Any number divided by itself is 1, so any fraction with the same numerator and denominator is equal to 1. Since multiplying any number by 1 results in the same number, multiplying any fraction by a fractional representation of the number 1 results in an equivalent fraction.

Example

Convert to three different equivalent fractions.

1.

2.

3.

If we were to divide each of the fractions above, we would find that they all equal 0.3.

Depending on the starting fraction, it is also possible to divide to find equivalent fractions. For example, dividing the numerator and denominator of by 4 would give us , another equivalent fraction of .

## Adding and subtracting fractions

Equivalent fractions are used when adding and subtracting fractions. In order to add or subtract a fraction, the fractions involved must be like fractions. If they are unlike fractions, then the unlike fractions must be converted into equivalent fractions that share the same denominator in order to be added or subtracted.

Examples

Convert the fractions in the following problems to equivalent fractions such that the fractions all share the same denominator, then add or subtract the fractions.

1. :

The least common multiple of 2 and 5 is 10, so we will convert the above fractions to equivalent fractions that have a denominator of 10 in order to add them.

2. :

The least common multiple of 7 and 21 is 21, so we only need to convert the first fraction to an equivalent fraction.